[Paper Review] Notes on entanglement entropy for excites holographic states in 2d

R. C. Rashkov|arXiv (Cornell University)|Jul 28, 2016

Quantum many-body systems6 references3 citations

TL;DR

This paper derives exact expressions for entanglement entropy contributions from excited holographic states in 2D CFTs using the replica trick, linking them to Aharonov invariants, Faber polynomials, Grunsky coefficients, and the tau-function of the dispersionless Toda hierarchy. The key result is a novel connection between entanglement entropy and integrable systems via the second derivatives of the dToda tau-function, suggesting deep links between holography, conformal field theory, and integrable hierarchies.

ABSTRACT

In this work we revisit the problem of contributions of excited holographic states to the entanglement entropy in two-dimensional conformal field theories. Using the results of replica trick method we find three expressions for these contributions. First, we express the contribution of the excited states in terms of Aharonov invariants. It is shown that beside the Schwarzian, the one-point functions of descendants of energy-momentum also contribute. Given Schwarz-Christoffel map, the contributions to any order can be easily computed. The second expression relates the entanglement entropy of excited states to Faber polynomials and Grunsky coefficients. Based on the relation of Grunsky coefficiens to tau-funcion of dispersionless Toda hierarchy, we find the third expression for contributions of excited holographic states to the entanglement entropy.

Motivation & Objective

• To understand how excited holographic states in 2D CFTs contribute to entanglement entropy beyond the vacuum case.
• To explore the role of conformal transformations and the replica trick in computing Rényi and entanglement entropy for non-vacuum states.
• To establish connections between entanglement entropy and mathematical structures such as Aharonov invariants, Faber polynomials, and Grunsky coefficients.
• To conjecture a fundamental relationship between entanglement entropy and the dispersionless Toda hierarchy through the tau-function.
• To provide a framework for computing entanglement entropy contributions to all orders using conformal mapping and integrable systems tools.

Proposed method

• Uses the replica trick to compute Rényi entropy and extract entanglement entropy in the n→1 limit for excited states.
• Expresses the entanglement entropy difference S_vac - S_ex in terms of the Schwarz-Christoffel map and Aharonov invariants, including pre-Schwarzian and Schwarzian derivatives.
• Relates the entanglement entropy to Faber polynomials and Grunsky coefficients via conformal mapping theory.
• Establishes that Grunsky coefficients are second derivatives of the free energy of the dispersionless Toda hierarchy.
• Derives a closed-form expression for the entanglement entropy difference using the dToda tau-function, showing it depends on ∂²F/∂tₘ∂tₙ.
• Demonstrates that the full entanglement entropy contribution can be computed order-by-order via the generating function of Grunsky coefficients.

Experimental results

Research questions

• RQ1 How do excited holographic states in 2D CFTs contribute to entanglement entropy beyond the vacuum state?
• RQ2 What is the role of the one-point function of the energy-momentum tensor and its descendants in determining entanglement entropy?
• RQ3 Can the entanglement entropy of excited states be systematically expanded using conformal mapping and special functions like Faber polynomials?
• RQ4 How are Grunsky coefficients related to the entanglement entropy in the context of 2D CFTs and holography?
• RQ5 Is there a deeper connection between entanglement entropy and integrable systems such as the dispersionless Toda hierarchy?

Key findings

• The entanglement entropy difference S_vac - S_ex is expressed as a logarithmic function of the ratio involving the Schwarz-Christoffel map and its derivatives, with contributions from both pre-Schwarzian and Schwarzian invariants.
• The contribution of excited states to entanglement entropy includes not only the one-point function of the stress tensor but also all descendants of the identity operator.
• The problem is reformulated using Faber polynomials and Grunsky coefficients, which provide a systematic way to compute entanglement entropy contributions to any order.
• Grunsky coefficients are shown to be second derivatives of the free energy of the dispersionless Toda hierarchy, establishing a direct link between entanglement entropy and integrable systems.
• The entanglement entropy difference is expressed as S_vac - S_ex = (c/12) log(1 + (z−w)² ∑_{m,n} ∂²F/∂tₘ∂tₙ z^{−m−1}w^{−n−1}), where F is the dToda free energy.
• The paper conjectures a profound connection between holographic entanglement entropy and the dispersionless Toda hierarchy, suggesting that entanglement entropy may be governed by integrable dynamics.